The numerical range of selfadjoint matrix polynomials

Abstract

The numerical range of a selfadjoint matrix polynomial $L\left( \lambda \right) = \sum\nolimits_j^\ell { = 0{\lambda ^j}{A_j}}$ is the set of points μ∈ℂ for which x* L(μ)x = 0 for some nonzero vector x. As for the classical eigenvalue problem (when L(λ) = λ I − A), the spectrum of L(λ) is contained in its numerical range. Properties of the numerical range are investigated with special emphasis on the cases when L(λ) has only real spectrum (and, possibly, the point at infinity) and when the coefficients of the matrix polynomial are real symmetric matrices.

Research supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.